\beginsection{2.2}

Show that $2^{1/3}$, $5^{1/7}$, and $(13)^{1/4}$ do not represent
rational numbers.

\medskip
$2^{1/3}$ is a solution to the polynomial $x^3-2=0$.
By the Rational Zeroes Theorem, the only possible rational solutions are
$\pm1$ and $\pm2$, neither of which is an actual solution.
Therefore, $2^{1/3}$ is not a rational number.

\medskip
$5^{1/7}$ is a solution to the polynomial $x^7-5=0$.
By the Rational Zeroes Theorem, the only possible rational solutions are
$\pm1$ and $\pm5$, neither of which is an actual solution.
Therefore, $5^{1/7}$ is not a rational number.

\medskip
$13^{1/4}$ is a solution to the polynomial $x^4-13=0$.
By the Rational Zeroes Theorem, the only possible rational solutions are
$\pm1$ and $\pm13$, neither of which is an actual a solution.
Therefore, $13^{1/4}$ is not a rational number.

